Guided Time-optimal Model Predictive Control of a Multi-rotor

TL;DR

This work tackles the time-optimal control problem for a multi-rotor by addressing under-actuation and nonlinear dynamics with a two-part approach: (1) thrust limit decomposition to convert the nonlinear thrust constraint into axis-aligned linear bounds, and (2) guided time-optimal MPC (GTOMPC) that uses a time-optimal guidance trajectory to steer a linear receding-horizon optimizer. The decomposition relies on balancing axis-wise flight times via an iterative algorithm based on Theorem I, producing an optimal distribution of thrust across the $x$, $y$, and $z$ axes. The GTOMPC framework then solves a quadratic cost over a finite horizon, penalizing deviations from the time-optimal trajectory while respecting acceleration and jerk bounds, enabling real-time operation. Experiments with 1000 random state pairs and outdoor hex-rotor flights show faster time-to-target and better utilization of thrust compared to decoupled time-optimal trajectory planning, validating both feasibility and practical impact.

Abstract

Time-optimal control of a multi-rotor remains an open problem due to the under-actuation and nonlinearity of its dynamics, which make it difficult to solve this problem directly. In this paper, the time-optimal control problem of the multi-rotor is studied. Firstly, a thrust limit optimal decomposition method is proposed, which can reasonably decompose the limited thrust into three directions according to the current state and the target state. As a result, the thrust limit constraint is decomposed as a linear constraint. With the linear constraint and decoupled dynamics, a time-optimal guidance trajectory can be obtained. Then, a cost function is defined based on the time-optimal guidance trajectory, which has a quadratic form and can be used to evaluate the time-optimal performance of the system outputs. Finally, based on the cost function, the time-optimal control problem is reformulated as an MPC (Model Predictive Control) problem. The experimental results demonstrate the feasibility and validity of the proposed methods.

Figures (6)

• Figure 1: (a) Independent acceleration bounds limit; (b) Dynamic acceleration bounds limit obtained through thrust limit decomposition
• Figure 2: Control architecture
• Figure 3: (a) $\boldsymbol{p}^*$ regarding to different $a_{\max}$ (dashed line regarding to the initial $a_{\max}=[5.774, 5.774, 5.774]^T (m/s^2)$ ; solid line regarding to the decomposed $a_{\max}=[9.028, 2.225, 3.612]^T (m/s^2)$ ); (b) convergence diagram of the direction of $a_{\max}$ and regarding $\boldsymbol{p}^*$ as the number of decomposition iteration increases (demonstrated in the $x$-$o$-$y$ plane; grey lines regarding 1-th $\sim$ 9-th iteration).
• Figure 4: The mean and standard deviation of $\rho _{t_{\delta}}$ and $\rho _{t_{\min}}$ of the thrust limit decomposition obtained over 1000 pairs of random data.
• Figure 5: The hex-rotor is controlled by the proposed method to fly from a line trajectory to switch into a circular trajectory.